Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 35, 2010, 255-274

ON LENGTH SPECTRUM METRICS AND WEAK METRICS ON TEICHMÜLLER SPACES OF SURFACES WITH BOUNDARY

Lixin Liu, Athanase Papadopoulos, Weixu Su and Guillaume Théret

Zhongshan University, Department of Mathematics
510275, Guangzhou, P.R. China; mcsllx 'at' mail.sysu.edu.cn

Université de Strasbourg and CNRS, Institut de Recherche Mathématique Avancée
7 rue René Descartes, 67084 Strasbourg Cedex, France; Max-Planck-Institut für Mathematik
Vivatsgasse 7, 53111 Bonn, Germany; papadopoulos 'at' math.u-strasbg.fr

Zhongshan University, Department of Mathematics
510275, Guangzhou, P.R. China; su023411040 'at' 163.com

Max-Planck-Institut für Mathematik
Vivatsgasse 7, 53111 Bonn, Germany; theret 'at' mpim-bonn.mpg.de

Abstract. We define and study natural metrics and weak metrics on the Teichmüller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of properly embedded arcs in the surface. We give a comparison between the defined metrics on regions of Teichmüller space which we call \varepsilon₀-relative \epsilon-thick parts, for \epsilon > 0 and \varepsilon₀ \geq \epsilon > 0. We compare the topologies defined by these metrics on Teichmüller space and we study divergence to infinity with respect to these various metrics.

2000 Mathematics Subject Classification: Primary 32G15, 30F30, 30F60.

Key words: Teichmüller space, length spectrum metric, length spectrum weak metric, Thurston's asymmetric metric.

Reference to this article: L. Liu, A. Papadopoulos, W. Su and G. Théret: On length spectrum metrics and weak metrics on Teichmüller spaces of surfaces with boundary. Ann. Acad. Sci. Fenn. Math. 35 (2010), 255-274.

Full document as PDF file

doi:10.5186/aasfm.2010.3515